| L(s) = 1 | + (−7.08 − 1.24i)2-s + (−191. − 69.8i)4-s + (−349. − 416. i)5-s + (22.4 − 8.16i)7-s + (2.86e3 + 1.65e3i)8-s + (1.95e3 + 3.38e3i)10-s + (1.54e4 − 1.84e4i)11-s + (−8.79e3 − 4.98e4i)13-s + (−169. + 29.8i)14-s + (2.18e4 + 1.82e4i)16-s + (3.75e4 − 2.16e4i)17-s + (−7.51e4 + 1.30e5i)19-s + (3.79e4 + 1.04e5i)20-s + (−1.32e5 + 1.11e5i)22-s + (1.08e5 − 2.99e5i)23-s + ⋯ |
| L(s) = 1 | + (−0.442 − 0.0780i)2-s + (−0.749 − 0.272i)4-s + (−0.559 − 0.666i)5-s + (0.00933 − 0.00339i)7-s + (0.700 + 0.404i)8-s + (0.195 + 0.338i)10-s + (1.05 − 1.25i)11-s + (−0.308 − 1.74i)13-s + (−0.00440 + 0.000775i)14-s + (0.332 + 0.279i)16-s + (0.449 − 0.259i)17-s + (−0.576 + 0.999i)19-s + (0.237 + 0.652i)20-s + (−0.566 + 0.475i)22-s + (0.389 − 1.06i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.382i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.924 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.106682 + 0.536957i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.106682 + 0.536957i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (7.08 + 1.24i)T + (240. + 87.5i)T^{2} \) |
| 5 | \( 1 + (349. + 416. i)T + (-6.78e4 + 3.84e5i)T^{2} \) |
| 7 | \( 1 + (-22.4 + 8.16i)T + (4.41e6 - 3.70e6i)T^{2} \) |
| 11 | \( 1 + (-1.54e4 + 1.84e4i)T + (-3.72e7 - 2.11e8i)T^{2} \) |
| 13 | \( 1 + (8.79e3 + 4.98e4i)T + (-7.66e8 + 2.78e8i)T^{2} \) |
| 17 | \( 1 + (-3.75e4 + 2.16e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (7.51e4 - 1.30e5i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-1.08e5 + 2.99e5i)T + (-5.99e10 - 5.03e10i)T^{2} \) |
| 29 | \( 1 + (9.45e5 + 1.66e5i)T + (4.70e11 + 1.71e11i)T^{2} \) |
| 31 | \( 1 + (-7.36e5 - 2.68e5i)T + (6.53e11 + 5.48e11i)T^{2} \) |
| 37 | \( 1 + (-1.21e6 - 2.10e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (2.52e6 - 4.45e5i)T + (7.50e12 - 2.73e12i)T^{2} \) |
| 43 | \( 1 + (2.24e6 + 1.88e6i)T + (2.02e12 + 1.15e13i)T^{2} \) |
| 47 | \( 1 + (-5.69e5 - 1.56e6i)T + (-1.82e13 + 1.53e13i)T^{2} \) |
| 53 | \( 1 + 3.88e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (6.87e6 + 8.19e6i)T + (-2.54e13 + 1.44e14i)T^{2} \) |
| 61 | \( 1 + (1.49e7 - 5.43e6i)T + (1.46e14 - 1.23e14i)T^{2} \) |
| 67 | \( 1 + (-1.45e6 - 8.24e6i)T + (-3.81e14 + 1.38e14i)T^{2} \) |
| 71 | \( 1 + (-9.42e6 + 5.43e6i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (-1.18e7 + 2.05e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (7.07e6 - 4.01e7i)T + (-1.42e15 - 5.18e14i)T^{2} \) |
| 83 | \( 1 + (-5.20e7 - 9.17e6i)T + (2.11e15 + 7.70e14i)T^{2} \) |
| 89 | \( 1 + (3.60e7 + 2.07e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (7.44e7 + 6.24e7i)T + (1.36e15 + 7.71e15i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.15280036999732254565875748533, −10.83485922042245705476885349013, −9.792069472028183021382918712827, −8.511030401464694199409299133324, −8.053347515045690213740577565410, −6.01667368607377590120423180563, −4.77699512288524685193107863447, −3.44132200183908447453073985622, −1.09617112742461339471684656445, −0.24437971338486044797789373975,
1.68564844186726041207103404537, 3.72164402079532204636477438043, 4.64839093635536155968666587159, 6.80276684188780638975962478302, 7.50883014205989099900239860689, 9.077527682326151817954667844914, 9.650460949284994425840954673505, 11.19428245737643766960539183673, 12.11014745894133238706000248822, 13.35068491423764537153150022838
